3.1365 \(\int \frac{(5-x) \sqrt{2+3 x^2}}{(3+2 x)^5} \, dx\)

Optimal. Leaf size=104 \[ -\frac{89 \left (3 x^2+2\right )^{3/2}}{2940 (2 x+3)^3}-\frac{13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}-\frac{33 (4-9 x) \sqrt{3 x^2+2}}{8575 (2 x+3)^2}-\frac{198 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]

[Out]

(-33*(4 - 9*x)*Sqrt[2 + 3*x^2])/(8575*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(140
*(3 + 2*x)^4) - (89*(2 + 3*x^2)^(3/2))/(2940*(3 + 2*x)^3) - (198*ArcTanh[(4 - 9*
x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8575*Sqrt[35])

_______________________________________________________________________________________

Rubi [A]  time = 0.153518, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{89 \left (3 x^2+2\right )^{3/2}}{2940 (2 x+3)^3}-\frac{13 \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}-\frac{33 (4-9 x) \sqrt{3 x^2+2}}{8575 (2 x+3)^2}-\frac{198 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{8575 \sqrt{35}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^5,x]

[Out]

(-33*(4 - 9*x)*Sqrt[2 + 3*x^2])/(8575*(3 + 2*x)^2) - (13*(2 + 3*x^2)^(3/2))/(140
*(3 + 2*x)^4) - (89*(2 + 3*x^2)^(3/2))/(2940*(3 + 2*x)^3) - (198*ArcTanh[(4 - 9*
x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(8575*Sqrt[35])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.1926, size = 99, normalized size = 0.95 \[ - \frac{33 \left (- 18 x + 8\right ) \sqrt{3 x^{2} + 2}}{17150 \left (2 x + 3\right )^{2}} - \frac{198 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{300125} - \frac{89 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{2940 \left (2 x + 3\right )^{3}} - \frac{13 \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{140 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**5,x)

[Out]

-33*(-18*x + 8)*sqrt(3*x**2 + 2)/(17150*(2*x + 3)**2) - 198*sqrt(35)*atanh(sqrt(
35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/300125 - 89*(3*x**2 + 2)**(3/2)/(2940*(2*x
+ 3)**3) - 13*(3*x**2 + 2)**(3/2)/(140*(2*x + 3)**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.140898, size = 85, normalized size = 0.82 \[ \frac{-1188 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \sqrt{3 x^2+2} \left (2217 x^3+10134 x^2-304 x+26028\right )}{(2 x+3)^4}+1188 \sqrt{35} \log (2 x+3)}{1800750} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^5,x]

[Out]

((-35*Sqrt[2 + 3*x^2]*(26028 - 304*x + 10134*x^2 + 2217*x^3))/(3 + 2*x)^4 + 1188
*Sqrt[35]*Log[3 + 2*x] - 1188*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2]
)])/1800750

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 149, normalized size = 1.4 \[ -{\frac{13}{2240} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{89}{23520} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{33}{17150} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{297}{300125} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{198}{300125}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}-{\frac{198\,\sqrt{35}}{300125}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{891\,x}{300125}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+2)^(1/2)/(2*x+3)^5,x)

[Out]

-13/2240/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(3/2)-89/23520/(x+3/2)^3*(3*(x+3/2)^2-
9*x-19/4)^(3/2)-33/17150/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(3/2)-297/300125/(x+3/
2)*(3*(x+3/2)^2-9*x-19/4)^(3/2)+198/300125*(12*(x+3/2)^2-36*x-19)^(1/2)-198/3001
25*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))+891/3001
25*x*(3*(x+3/2)^2-9*x-19/4)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.783532, size = 200, normalized size = 1.92 \[ \frac{198}{300125} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{99}{17150} \, \sqrt{3 \, x^{2} + 2} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{89 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{2940 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{66 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}{8575 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{297 \, \sqrt{3 \, x^{2} + 2}}{17150 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(x - 5)/(2*x + 3)^5,x, algorithm="maxima")

[Out]

198/300125*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3
)) + 99/17150*sqrt(3*x^2 + 2) - 13/140*(3*x^2 + 2)^(3/2)/(16*x^4 + 96*x^3 + 216*
x^2 + 216*x + 81) - 89/2940*(3*x^2 + 2)^(3/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 66/
8575*(3*x^2 + 2)^(3/2)/(4*x^2 + 12*x + 9) - 297/17150*sqrt(3*x^2 + 2)/(2*x + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.289282, size = 167, normalized size = 1.61 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (2217 \, x^{3} + 10134 \, x^{2} - 304 \, x + 26028\right )} \sqrt{3 \, x^{2} + 2} - 594 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{1800750 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(x - 5)/(2*x + 3)^5,x, algorithm="fricas")

[Out]

-1/1800750*sqrt(35)*(sqrt(35)*(2217*x^3 + 10134*x^2 - 304*x + 26028)*sqrt(3*x^2
+ 2) - 594*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(sqrt(35)*(93*x^2 - 36*
x + 43) + 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(16*x^4 + 96*x^3 +
216*x^2 + 216*x + 81)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+2)**(1/2)/(3+2*x)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.31848, size = 244, normalized size = 2.35 \[ \frac{1}{9604000} \, \sqrt{35}{\left (739 \, \sqrt{35} \sqrt{3} + 6336 \,{\rm ln}\left (\sqrt{35} \sqrt{3} - 9\right )\right )}{\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{198}{300125} \, \sqrt{35}{\rm ln}\left (\sqrt{35}{\left (\sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac{\sqrt{35}}{2 \, x + 3}\right )} - 9\right ){\rm sign}\left (\frac{1}{2 \, x + 3}\right ) - \frac{1}{823200} \,{\left (\frac{35 \,{\left (\frac{7 \,{\left (\frac{1365 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )}{2 \, x + 3} - 257 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 9 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 2217 \,{\rm sign}\left (\frac{1}{2 \, x + 3}\right )\right )} \sqrt{-\frac{18}{2 \, x + 3} + \frac{35}{{\left (2 \, x + 3\right )}^{2}} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 2)*(x - 5)/(2*x + 3)^5,x, algorithm="giac")

[Out]

1/9604000*sqrt(35)*(739*sqrt(35)*sqrt(3) + 6336*ln(sqrt(35)*sqrt(3) - 9))*sign(1
/(2*x + 3)) - 198/300125*sqrt(35)*ln(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)
^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sign(1/(2*x + 3)) - 1/823200*(35*(7*(1365*sig
n(1/(2*x + 3))/(2*x + 3) - 257*sign(1/(2*x + 3)))/(2*x + 3) + 9*sign(1/(2*x + 3)
))/(2*x + 3) + 2217*sign(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)